Superposition & Standing Waves Superposition Principle - - PDF document

Superposition & Standing Waves

• Superposition Principle
• Interference of Waves
• Standing Waves
• Homework

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Superposition Principle

• Overlapping waves algebraically add to pro-

duce a resultant wave (or net wave).

• Overlapping waves do not in any way alter

the travel of each other.

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Superposition Principle (cont’d)

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Interference of Waves Consider two sinusoidal waves that are travel- ing in the positive x-direction and have the same frequency, wavelength and amplitude but differ in phase y1 = A sin (kx − ωt) y2 = A sin (kx − ωt + φ) y = y1+y2 = A [sin (kx − ωt) + sin (kx − ωt + φ)] sin a + sin b = 2 sin

    

a + b 2

     cos     

a − b 2

    

y =

    2A cos φ

2

     sin     kx − ωt + φ

2

    

When φ = 0, 2π, 4π, . . . then 2A cos φ 2 = ±2A ⇒ constructive interference When φ = π, 3π, . . . then 2A cos φ 2 = 0 ⇒ destructive interference

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Interference of Waves (cont’d)

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Example Two identical sinusoidal waves, moving in the same direction along a stretched string, inter- fere with each other. The amplitude of each wave is 9.8 mm, and the phase difference be- tween them is 100◦. (a) What is the amplitude

• f the resultant wave due to the interference of

these two waves, and what type of interference

• ccurs? (b) What phase difference, in radians

and wavelengths, will give the resultant wave an amplitude of 4.9 mm?

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Example Solution (a) A′ = 2A cos 1 2φ = 2 (9.8 mm) cos 100◦ 2 = 13 mm (b) A′ = 2A cos 1 2φ φ = 2 cos−1 A′ 2A φ = 2 cos−1 4.9 mm 2 (9.8 mm) = ±2.6 rad φ = ±2.6 rad

    

1 wavelength 2π rad

    

φ = ±0.41 wavelengths

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Standing Waves Consider two waves with the same amplitude, wavelength, and frequency traveling in oppo- site directions y1 = A sin (kx − ωt) y2 = A sin (kx + ωt) y = y1+y2 = A [sin (kx − ωt) + sin (kx + ωt)] sin a + sin b = 2 sin

    

a + b 2

     cos     

a − b 2

    

y = (2A sin kx) cos ωt y = 0 when x = nλ 2 where n = 0, 1, 2, . . . ⇒ nodes y = 2A cos ωt when x =

    n + 1

2

    

λ 2 where n = 0, 1, 2, . . . ⇒ antinodes

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Standing Waves (cont’d)

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Homework Set 25 - Due Fri. Nov. 12

• Read Sections 14.1-14.4
• Answer Questions 14.1 & 14.2
• Do Problems 14.1, 14.5, 14.8 & 14.10

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